We explore the structure of nuclei and topological defects
during the phase transition in lyotropic chromonic liquid
crystals (LCLCs). The LCLCs are formed by self-assembled
molecular aggregates and show a broad biphasic region. The
defects emerge as a result of two mechanisms. 1) Surface
anisotropy mechanism that endows each N nucleus (tactoid)
with topological defects thanks to tangential orientation of
the director at the interface, and 2) Kibble mechanism with
defects forming when differently oriented tactoids.
Different scenarios of phase transition involve positive (N-
in-I) and negative (I-in-N) tactoids with non-trivial
topology of the director field and also multiply connected
tactoids-in-tactoids configurations. The interface limiting
a tactoid shows a certain number of cusps. The each cusp
contains a point defect-boojum. The number of cusps shows
how many times the director becomes perpendicular to the
interface. We derive conservation laws that connect the
number of cusps to the topological strength of defects in
the N part of the simply- and multiply-connected tactoids.
We demonstrate how the elastic anisotropy of the N phase
results in non-circular shape of the disclination cores. A
generalized Wulff construction is used to derive the shape
of tactoids as the function of interfacial tension
anisotropy in the frozen director. The shapes and structures
of tactoids and topological defects demonstrate an important
role of surface anisotropy in morphogenesis of phase
transitions in LCs.